Let \(X_m^{(b)}(\theta ) = \sqrt{m}(G_m^{(b)}(\theta )-G_n(\theta )\) , by Gaussian re-weighting it is a Gaussian process. We will use an inequality for separable sub-Gaussian processes, specifically Corollary 2.2.8 in [1]}, to bound \(M_n\) . The following verifies the assumptions required to apply the result. For any \(\theta _1,\theta _2 \in \Theta \) , we have:
\(&X_m^{(b)}(\theta _1)- X_m^{(b)}(\theta _2) = \frac{1}{\sqrt{m}} \sum _{i=1}^m (w_i^{b}-1) [ \partial _\theta q(z_i,\theta _1)-\partial _\theta q(z_i,\theta _2) ] \sim \mathcal {N}\left(0,V_m(\theta _1,\theta _2) \right),\) 
